Prerequisites & Notation

Before You Begin

This chapter starts the optimization thread of the book. Every subsequent chapter (6–8, 10–13) will design variants of the algorithms introduced here. Make sure the prerequisites below are solid before proceeding — particularly the convex-optimization tools, which are used without re-derivation.

Classical MIMO beamforming: MRC, ZF, MMSE(Review ch15)
Self-check: For a MISO channel $\mathbf{h}$ , can you derive the MRT beamformer $\mathbf{v} = \mathbf{h}/\|\mathbf{h}\|$ and state its optimality?
Convex optimization: Lagrangian, KKT conditions, strong duality(Review ch03)
Self-check: Can you state the KKT conditions for $\min_{\mathbf{x}} f(\mathbf{x})$ s.t. $g(\mathbf{x}) \leq 0$ ?
Matrix calculus: gradient and Hessian of quadratic forms(Review ch01)
Self-check: What is $\nabla_{\mathbf{v}} (\mathbf{v}^{H} \mathbf{A} \mathbf{v})$ ?
Block coordinate descent convergence theorems
Self-check: Under what conditions does block coordinate descent converge to a stationary point?
The cascaded channel model from Chapter 3(Review ch03)
Self-check: Write the effective channel $\mathbf{h}_{\text{eff}}^H$ in terms of $\mathbf{h}_d, \mathbf{h}_2, \boldsymbol{\Phi}, \mathbf{H}_1$ .

Notation for This Chapter

Optimization-specific notation. The core RIS symbols carry over from Chapters 1–3; here we add the objective-function notation and iteration indices.

Symbol	Meaning	Introduced
$\mathbf{W}$	Active precoding matrix, $\mathbf{W} \in \mathbb{C}^{N_t \times K}$	s01
$\mathbf{v}_{k}$	$k$ -th column of $\mathbf{W}$ : the beamforming vector for user $k$	s01
$R_k(\mathbf{W}, \boldsymbol{\Phi})$	Achievable rate for user $k$ as a function of both beamformers	s01
$f(\mathbf{W}, \boldsymbol{\Phi})$	Objective function (e.g., sum rate, max-min rate)	s01
$\mathbf{W}^{(i)}$ , $\boldsymbol{\Phi}^{(i)}$	Iterates at alternating-optimization step $i$	s02
$\mathcal{F}_{\text{active}}$	Feasible set for active beamforming: $\{\mathbf{W} : \text{tr}(\mathbf{W}^{H}\mathbf{W}) \leq P_t\}$ (convex)	s03
$\mathcal{F}_{\text{passive}}$	Feasible set for passive beamforming: $\{\boldsymbol{\phi} : \|\phi_n\| = 1\}$ (non-convex torus)	s04

← Ch 4 The Joint Beamforming Problem